Hurwitz Groups of Large Rank

Deformation theory and finite simple quotients of triangle groups I

“…This is for the positive side. On the negative side we essentially prove a conjecture of Marion proposed in [21]: in that paper he studied (a, b, c)-generation of finite quasisimple (2,4,6), (2,6,6), (2, 6, 10) (3,4,4), (3,6,6), (4,6,12) 5,7,8,9,10,11,13,15,16,17,19,22,23,25,29,31, 37, 43} (r ≥ 4) (2,3,8) r ∈ {4, 5, 7, 9, 10, 11, 13, 17, 19, 25} (2,3,9) r ∈ {4, 5, 7, 10, 11, 13, 19} (2,3,10) r ∈ {4, 5, 7, 11, 13} (2,3,11) r ∈ {4, 5, 7, 13} (2,3,12) r ∈ {4, 5, 7, 13} (2, 3, c), c ≥ 13 r ∈ {4, 5, 7} (2,4,…”

“…We therefore do not have a good control on the ring of definition of ρ. As a result, we cannot give an explicit upper bound for p 0 or e in Definition 1.1, and so our method cannot give a result of the kind proved in [19] stating that for r ≥ 286 every finite simple group of type A r is a quotient of T 2,3,7 .…”

Hurwitz Groups of Intermediate Rank

“…For example, Lucchini, Tamburini and Wilson proved the following theorem [7,Corollary 1]. Theorem 1.1 (see [7]). (1) For each prime power q and each integer n 287, the group SL n (q) is a Hurwitz group.…”

“…We just mention a recent survey [10], where an overview of the known results is given. Particular attention is paid to the case of classical groups over various rings, especially over finite fields or the ring Z of integers; see [3,6,7]. As is shown in [3], many linear classical groups of rank less than 18 are not Hurwitz.…”

“…On the other hand, for all sufficiently large ranks, the groups SL n (q), Sp 2n (q), SU 2n (q) and + 2n (q) for any prime power q, and SU 2n+1 (q) and 2n+1 (q) for any odd prime power q, are known to be Hurwitz; see [6,7]. For example, Lucchini, Tamburini and Wilson proved the following theorem [7,Corollary 1]. Theorem 1.1 (see [7]).…”